arXiv subjects

Kübra Benli

Publications and source records attributed to Kübra Benli.

7 recordsLinked to original sources

arXiv2026

On the digits of the sum of proper divisors

We study several probabilistic questions concerning the digits of $s(n)$, the sum of proper divisors of an integer $n$. In particular, we show that $s(n)$ obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function $k(x) \rightarrow \infty$, almost all integers $n \leq x$ have every decimal digit occurring among the first $k(x)$ digits and the last $k(x)$ digits of $s(n)$. We also present an upper bound for the number of composite integers $n$ up to $x$ for which $s(n)$ is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs $n$ were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set $s^{-1}(\mathcal{A})$, where $\mathcal{A}$ is a set of integers with missing digits. Our result for composite $n$ shows that the count is much smaller when prime inputs are excluded.

math.NT↗

arXiv2026

Explicit Deuring-Heilbronn phenomenon for Dirichlet $L$-functions

Assuming the existence of a Landau-Siegel zero, we establish an explicit Deuring-Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip, and improves over the previous best known explicit estimate due to Thorner and Zaman.

math.NT↗

arXiv2025

Joint distributions of error terms for primes in arithmetic progressions modulo 11

We provide a formula for the logarithmic density of the set of positive real numbers on which two prime counting functions $ψ(x;q,a)$ and $ψ(x;q,b)$ are simultaneously larger than their asymptotic main terms, as well as a method for calculating the numerical values of such densities with rigorously bounded errors. We apply these formulas to the pairwise races in the case $q=11$, determining which pairs of residues $a$ and $b$ are more or less correlated in this way. The outcomes when $q=11$ provide a deeper mathematical illumination of the "mirror image" and "cyclic ordering" phenomena observed by Bays and Hudson.

math.NT↗

arXiv2024

An annotated bibliography for comparative prime number theory

The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory (through mid-2024) together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.

math.NT↗

arXiv2023

A discrete mean value of the Riemann zeta function

In this work, we estimate the sum \begin{align*} \sum_{0 < \Im(ρ) \leq T} ζ(ρ+α)X(ρ) Y(1\!-\! ρ) \end{align*} over the nontirival zeros $ρ$ of the Riemann zeta funtion where $α$ is a complex number with $α\ll 1/\log T$ and $X(\cdot)$ and $Y(\cdot)$ are some Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where $ζ(ρ+α)$ is replaced by $ζ^{(m)}(ρ)$ for all $m\in\mathbb{N}$. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann Hypothesis we obtain the lower bound \begin{align*} \sum_{0 < \Im(ρ) < T} | ζ^{(m)}(ρ)|^{2k} \gg T(\log T)^{k^2+2km+1} \quad \quad (k,m\in\mathbb{N}) \end{align*} which was previously known under the Generalized Riemann Hypothesis, in the case $m=1$.

math.NT↗

arXiv2023

Sums of proper divisors with missing digits

Let $s(n)$ denote the sum of proper divisors of an integer $n$. In 1992, Erdős, Granville, Pomerance, and Spiro (EGPS) conjectured that if $\mathcal{A}$ is a set of integers with asymptotic density zero then $s^{-1}(\mathcal{A})$ also has asymptotic density zero. In this paper we show that the EGPS conjecture holds when $\mathcal{A}$ is taken to be a set of integers with missing digits. In particular, we give a sharp upper bound for the size of this preimage set. We also provide an overview of progress towards the EGPS conjecture and survey recent work on sets of integers with missing digits.

math.NT↗

arXiv2017

Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach

Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $ε> 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(ε)$, the number of prime cubic residues $q < p^{1/2+ε}$ exceeds $p^{ε/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{1/9}$ prime quadratic residues $q<p$.

math.NT↗